Bribery of the permit-issuers has been an issue at least since the Knapp Commission investigated NYPD corruption in the 1970s. As I recall, that led to moving the decision making from officials in the precincts to officials at headquarters, in the hopes that would make corruption a little more difficult. There was probably undocumented corruption since the system began--you can find news articles from the 1920s and 1930s about mobsters holding pistol carry permits.

UPDATE: this fellow paid $12,500 and got a pistol carry permit, despite having a felony record. He was shocked to find that he still couldn't buy a gun, since he flunked the NICS background check.

re: you can find news articles from the 1920s and 1930s about mobsters holding pistol carry permits.

Actually the Sullivan Act which was passed in the early 1900's was named after I giy named Sullivan who was on the payroll of one of the major NYC gangs. The law was used to basically allow the gang members of that gang while "disarming" the rival gang's members. It also meant that the public was at a disadvantage with respect to the gang members. You wouldn't want the public able to defend itself from the strong arm of the mob, would ya. NYC was an open cess pool of gangs with police on the payroll in general and a good number of areas that nobody would go if they did not live there. West Side Story is a really sanitized version of the reality. It was not called Hell's Kitchen for nothing. Then as now if you were connected you got a "permit" if not you did not exist.